hi please someone describe me why we use slip boundary condition in free surface problems. thanks

I think the boundary is a streamline in both cases. They both have no tagent traction if the free surface has uniform tension. However, when you solve the real free surface, you don't use slip boundary. You use a stress balance equation for free surface, and the stress comes from outside pressure and surface tension.

Could someone please elaborate on this slip BC condition and its implementation.

At a standard wall boundary we use a no slip condition, so the velocity would be zero at the wall. If this condition was used for free surface flows it would mean that the interface (gas-liquid) can not move in the vacinity of the wall. In reality this is true but I believe the mechanism differs in that the liquid molecules are so small that they simply move toward or away from the wall as required (as the interface moves along the wall)and therby creating the "illusion?", on a macroscopic scale, that the interface is slipping along the wall.

Slip BC boundary means that in the normal direction, you can't have velocity, and in the tangential direction, you don't have stress (t.(n.T)=0). I use this boundary condition to get the result which is used as initial guess for my free surface problem.